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Preinstructional Planning

Objectives

Students will:

• Define the term unit rate
• Learn how to calculate unit rates
• Solve multistep ratio and unit rate problems

Materials

Set Up

• Make class sets of the Designing RATIOnally Worksheet: A First-Rate Design printable.
• Print a copy of the Answer Key: Designing RATIOnally Worksheets printable for your own use.
• Optional: If you are planning to use the Ratio Design Challenge Digital Interactive Tool in class, set up the program ahead of time.
• Optional: Make copies of the Designing RATIOnally Worksheet: Keeping It All in Proportion printable to assign to advanced students after the lesson.

Lesson Directions

Introduction to New Material

Step 1:  Indicate that, after many years of paying high prices for his favorite drink at his local coffee bar, Tommy discovered that he could make the same “delicious” drink at home for a lot less money. His favorite chocolate syrup, Econo-Choc, comes in two sizes: a 2.5-liter container for \$16.25 and a 4-liter container for \$24.80. Which size would be the better buy for Tommy? Because the sizes of the two containers are different, it is difficult to easily determine which one is the better buy. To make the comparison easier, Tommy can calculate a unit rate for each container and compare. A unit rate is a ratio comparing two measures in which one of the measures has a value of 1 (the word “unit” in “unit rate” refers to one unit).

Step 2: Indicate that to compare the two sizes, a type of unit rate, the unit price, must be determined for each container of Econo-Choc by dividing the price by the size. This will yield a simplified price per one unit (in this case, one liter). When we convert two fractions so they have the same denominator, they can be easily compared. In other words, to compare the two differently sized containers, a price for the same amount within each container (in this case, one liter) must be calculated.

To do so, begin by expressing price-per-size as a ratio in fraction notation: \$16.25/2.5 liters. Then, divide as indicated by the fraction in order to simplify to a denominator of one. \$16.25 ÷ 2.5 = \$6.50/1 liter, more commonly said as \$6.50 per liter. For the second container: \$24.80/4 liters. \$24.80 ÷ 4 = \$6.20 per liter. Since the 4-liter container is \$0.30 cheaper per liter, it is the better buy (as long as Tommy is sure he will use all four liters before it goes bad).

Step 3: Indicate that calculating unit rates can be useful for other units of measure as well:

• If Tommy decides to start power walking to work off the extra calories he consumed by drinking his favorite beverage, and he walks 3 miles in 45 minutes, his unit rate is 4 miles per hour (3 miles ÷ 3/4 of an hour = 4). Indicate that the unit rate could also be depicted as 15 minutes per mile.
• To compare the speeds of two cars, compare miles per hour. That is, the number of miles the car traveled in one hour. If one car traveled 150 miles in 3 hours and another traveled 120 miles in 2 hours, compare their speeds by setting up the fractions 150 miles/3 hours and 120 miles/2 hours. Divide to find that the first car’s rate of speed was 50 miles in 1 hour (50 miles per hour) and the second car’s rate was 60 miles in 1 hour (60 miles per hour). The second car had the faster speed.

Step 4: Ask the class to provide examples of other rates they may be familiar with, e.g., miles per hour, miles per gallon of gas, dollars per hour (pay rates), cost per pound (produce), minutes per class period, etc.

Step 5: Ask the class to find the answer to the following problem:

Tommy is considering buying two different cars. On test drives, one car used 1.5 gallons of gas and traveled 48 miles. The other car used 1.7 gallons on a 51-mile drive. Which car gets better gas mileage (travels more miles per gallon of gas)?

Answer: The first car. To find miles per gallon, divide miles driven by gallons used. In this case 48/1.5 = 32. For the second car, 51/1.7 = 30, so the first car has better mileage per gallon.

Guided Practice

Step 6: Have students use the Ratio Design Challenge Digital Interactive Tool to practice what they have learned, or distribute the practice worksheet.

#### USING THE DIGITAL INTERACTIVE TOOL*

Select the “Aquarium Research Center” venue theme module on unit rate in the Ratio Design Challenge. As a class, complete the first two problems.

*The Digital Interactive Tool requires Internet access and either an interactive whiteboard or a computer/projector hookup.

#### USING THE DESIGNING RATIONALLY WORKSHEET: A FIRST-RATE DESIGN PRINTABLE

As a class, complete the first two problems of the Designing RATIOnally Worksheet: A First-Rate Design printable. Have students explain their reasoning and methods for coming up with the answers.

Note: This worksheet can also be assigned for homework if the online tool was used in class.

Step 7: Checking for Understanding: Address any student misconceptions as they occur. If the online tool was used, discuss the critical-thinking questions included at the end as a class. Wrap up by having several students each volunteer one fact they learned, share something they’d like to learn more about, or ask a question.

Independent Practice

Step 8: Assign students to complete additional practice using the Ratio Design Challenge Digital Interactive Tool, or have students complete the remainder of the Designing RATIOnally Worksheet: A First-Rate Design printable.

#### USING THE DIGITAL INTERACTIVE TOOL*

Assign small groups to complete the “Aquarium Research Center” venue theme module in the Ratio Design Challenge, which was begun during Guided Practice.

*The Digital Interactive Tool requires Internet access and either an interactive whiteboard or a computer/projector hookup.

#### USING THE DESIGNING RATIONALLY WORKSHEET: A FIRST-RATE DESIGN PRINTABLE

Assign the remaining problems in the Designing RATIOnally Worksheet: A First-Rate Design printable for students to complete individually.

Note: This worksheet can also be assigned for homework if the online tool was used in class.

Step 9: Checking for Understanding: Have students report their findings/answers to the class and share their conclusions. If the online tool was used, discuss the critical-thinking questions included at the end as a class. Address any misconceptions yourself or via student volunteers.

Supporting All Learners

For homework, assign the Designing RATIOnally Worksheet: Keeping It All in Proportion printable. This activity sheet contains multistep ratio and percentage problems designed to challenge students’ thinking while reinforcing lesson objectives on ratios and unit rate.

Lesson Extensions

Venue Design Activity

Once your students are familiar with ratio concepts and have had a chance to explore the online Ratio Design Challenge, consider assigning a project that gives students a chance to exercise their imagination and extend their skills and knowledge of ratios by designing a venue of their choice. After determining the type of venue, e.g., a state-of-the-art recording studio, students create a scale drawing of the venue. Students determine the scale of the drawing, i.e., the ratio between the size of an object in the drawing and its size in the real world. Students should select at least five items in their drawing and, taking into account their measurements and the drawing’s scale, calculate the size of the items in the real world.

Post Instructional

Standards

CCSS and NCTM:

• Grade 7: Unit rate with ratios of fractions; proportional relationships; multistep ratio and percent problems (CCSS 7.RP.1, 7.RP.2, 7.RP.3)
• Grades 6–8: Making sense of problems, reasoning, constructing an argument, modeling, using appropriate tools strategically, and attending to precision (CCSS Practice MP1-6); NCTM Number and Operations